Cantor and Infinity
Numbers and Infinite Numbers

We encounter problems when we start asking
questions like:
− What is a number?
− What is in...
Some Definitions

A set is a collection of well-defined, well-
distinguished objects. These objects are then
called the e...
Galileo’s Paradox of Equinumerosity

Consider the set of natural numbers Ν = {1, 2,
3, 4, …} and the set of perfect squar...
A Contradiction?
1.While some natural numbers are perfect
squares, some are clearly not. Hence the set N
must be more nume...
Many Contradictions?

We could repeat this reasoning for N and:
− The even numbers E = {2, 4, 6, …}
− Triples
− Cubes
− E...
One-to-One Correspondence

Galileo’s exact matching of the naturals with
the perfect squares constitutes an early use of
...
Galileo's 'Solution'

To resolve the paradox, Galileo concluded that
the concepts of “less,” “equal,” and “greater”
were ...
The Basis of Cantor's Theory

A bijection is a function giving an exact pairing
of the elements of two sets.

Two sets A...
Which sets have the same cardinality as N?

The cardinality of the natural numbers, |N|, is
usually written as

Other we...
Showing N~Q

We can establish a
one-to-one
correspondence
between the naturals
and the rationals.

We use an infinite 2-...
Are there any infinite sets with greater
cardinality than N?

We can show that |R| is greater |N|.

We cannot establish ...
Cantor's Diagonal Argument

For any
hypothesised
enumeration of the
real numbers, we
can show that there
is a real which ...
Transfinite Arithmetic

Cantor devised a
new type of
arithmetic for these
infinite numbers.

For the infinite
numbers we...
The Continuum Hypothesis

We have seen that |R| is greater than |N|.

But, are there any infinite numbers in between
the...
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Matthew infinitypresentation

  1. 1. Cantor and Infinity
  2. 2. Numbers and Infinite Numbers  We encounter problems when we start asking questions like: − What is a number? − What is infinity? − Is infinity a number? − If it is, can there be many infinite numbers?  We can resolve some aspects of these questions with a few simple ideas, rigorously applied.
  3. 3. Some Definitions  A set is a collection of well-defined, well- distinguished objects. These objects are then called the elements of the set.  For a given set S, the number of elements of S, denoted by |S|, is called the cardinal number, or cardinality, of S.  A set is called finite if its cardinality is a finite nonnegative integer.  Otherwise, the set is said to be infinite.
  4. 4. Galileo’s Paradox of Equinumerosity  Consider the set of natural numbers Ν = {1, 2, 3, 4, …} and the set of perfect squares (i.e. the squares of the naturals) S = {1, 4, 9, 16, 25, …}.  Galileo produced the following contradictory statements regarding these two sets ...
  5. 5. A Contradiction? 1.While some natural numbers are perfect squares, some are clearly not. Hence the set N must be more numerous than the set S, or |N| > |S|. 2.Since for every perfect square there is exactly one natural that is its square root, and for every natural there is exactly one perfect square, it follows that S and N are equinumerous, or |N| = |S|.
  6. 6. Many Contradictions?  We could repeat this reasoning for N and: − The even numbers E = {2, 4, 6, …} − Triples − Cubes − Etc.  In each case we have systematically picked out an infinite proper subset of N.  A is a proper subset of B if A is contained in B, but A ≠ B.
  7. 7. One-to-One Correspondence  Galileo’s exact matching of the naturals with the perfect squares constitutes an early use of a one-to-one correspondence between sets – the conceptual basis for Cantor’s approach to infinity.
  8. 8. Galileo's 'Solution'  To resolve the paradox, Galileo concluded that the concepts of “less,” “equal,” and “greater” were inapplicable to the cardinalities of infinite sets such as S and N, and could only be applied to finite sets.  Cantor showed how these concepts could be applied consistently, in a theory of the properties of infinite sets.
  9. 9. The Basis of Cantor's Theory  A bijection is a function giving an exact pairing of the elements of two sets.  Two sets A and B are said to be in a one-to- one correspondence if and only if there exists a bijection between the two sets. We then write A~B.  A set is said to be infinite if and only if it can be placed in a one-to-one correspondence with a proper subset of itself.
  10. 10. Which sets have the same cardinality as N?  The cardinality of the natural numbers, |N|, is usually written as  Other well-known infinite sets have cardinality − The integers, I, { …, -3, -2, -1, 0, 1, 2, 3, …} (fairly easy to show N~I). − The rationals, Q (a bit more difficult to show N~Q). Showing N~I
  11. 11. Showing N~Q  We can establish a one-to-one correspondence between the naturals and the rationals.  We use an infinite 2- d arrangement of Q, and a systematic path through this arrangement.
  12. 12. Are there any infinite sets with greater cardinality than N?  We can show that |R| is greater |N|.  We cannot establish a one-to-one correspondence between N and R.  |R| is a higher order of infinity, usually written as  Let's prove this ...
  13. 13. Cantor's Diagonal Argument  For any hypothesised enumeration of the real numbers, we can show that there is a real which is not in that enumeration.  We rely on forming a new real by the systematic alteration of the digits in the enumeration.
  14. 14. Transfinite Arithmetic  Cantor devised a new type of arithmetic for these infinite numbers.  For the infinite numbers we have rules such as these ... Note: the cardinality of the set of real numbers is often written 'c' for 'continuum'.
  15. 15. The Continuum Hypothesis  We have seen that |R| is greater than |N|.  But, are there any infinite numbers in between these?  The hypothesis that there is no infinite number between |N| and |R| is called the continuum hypothesis.  It was shown to be formally undecidable by Gödel and Cohen.

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